This document describes a model for populations that can change over time through people joining or leaving. The model represents population changes with a recurrence relation of S(n+1) = TS(n) + B, where S is the population state, T is a transition matrix describing how people move between groups, and B accounts for people joining or leaving. An example is given of a gym with yoga, crossfit, and aerobics programs where the populations change monthly based on people switching programs or joining/leaving. The initial populations, transition matrices, and B vectors are defined and the populations after 1 and 2 months are calculated.

2. Context
 In all of the examples we have seen so far, the
populations are fixed and no one can leave or enter, all
they can do is move around between nodes
 However, in reality, we can come and go as we please,
we are not trapped in a bubble

3. The equation
 We can represent a transition scenario with people
joining or leaving the population as:
𝑆 𝑛+1 = 𝑇𝑆 𝑛 + 𝐵
Where:
𝑆 𝑛+1 is the next state (a column matrix)
𝑆 𝑛 is the current state (a column matrix)
𝑇 is the transition matrix (a square matrix)
𝐵 is the addition or loss (a column matrix)
𝑆0 is the initial state

4. Example
 A gym has three programs: Yoga, CrossFit and Aerobics.
Initially, there are 100 people doing the Yoga Program, 200
people are in the CrossFit Program and 300 people in the
Aerobics program.
 Each month, 2% of people in Yoga change to CrossFit, 4%
of people in Yoga change to Aerobics and 3 new people join
the gym and pick Yoga
 Each month, 5% of people in CrossFit change to Yoga , 1%
of people in CrossFit change to Aerobics and 1 person
doing CrossFit leaves the gym all together
 Each month, 2% of people in Aerobics change to Yoga, 2%
of people in Aerobics change to CrossFit and 2 new people
join the gym and pick Aerobics

5. Example Continued
a) Set up a recurrence relation 𝑆 𝑛+1 = 𝑇𝑆 𝑛 + 𝐵
for the scenario by stating T and B.
b) State the initial state matrix
c) Find the number of people doing each activity after
one and two weeks.

6. Example cont.
a) Set up a recurrence relation 𝑆 𝑛+1 = 𝑇𝑆 𝑛 + 𝐵
for the scenario by stating T and B.
Each month, 2% of people in Yoga change to CrossFit, 4% of people in Yoga
change to Aerobics and 3 new people join the gym and pick Yoga
𝑌 𝐶 𝐴
𝑇 =
𝑌
𝐶
𝐴
0.94
0.02
0.04
𝐵 =
𝑌
𝐶
𝐴
3
Since each month 2% leave for CrossFit and
4% leave for Aerobics, there is 94% left over
who stay with Yoga.
Three people join Yoga, so we have positive
three in B.

7. Example cont.
a) Set up a recurrence relation 𝑆 𝑛+1 = 𝑇𝑆 𝑛 + 𝐵
for the scenario by stating T and B.
Each month, 5% of people in CrossFit change to Yoga , 1% of people in
CrossFit change to Aerobics and 1 person doing CrossFit leaves the gym all
together
𝑌 𝐶 𝐴
𝑇 =
𝑌
𝐶
𝐴
0.94 0.05
0.02 0.94
0.04 0.01
𝐵 =
𝑌
𝐶
𝐴
3
−1
Since each month 5% leave for Yoga and 4%
leave for Aerobics, there is 94% left who stay
with CrossFit.
Since each month one person leaves the gym
who is doing CrossFit we have -1.

8. Example cont.
a) Set up a recurrence relation 𝑆 𝑛+1 = 𝑇𝑆 𝑛 + 𝐵
for the scenario by stating T and B.
Each month, 2% of people in Aerobics change to Yoga, 2% of people in
Aerobics change to CrossFit and 2 new people join the gym and pick Aerobics
𝑌 𝐶 𝐴
𝑇 =
𝑌
𝐶
𝐴
0.94 0.05 0.02
0.02 0.94 0.02
0.04 0.01 0.96
𝐵 =
𝑌
𝐶
𝐴
3
−1
2
Since each month 2% leave for Yoga and 2%
leave for CrossFit, there is 96% left who stay
with Aerobics.
Two new people join and do Aerobics, so we
have +2.

9. Example cont.
a) Set up a recurrence relation 𝑆 𝑛+1 = 𝑇𝑆 𝑛 + 𝐵
for the scenario by stating T and B.
𝑌 𝐶 𝐴
𝑆 𝑛+1 =
𝑌
𝐶
𝐴
0.94 0.05 0.02
0.02 0.94 0.02
0.04 0.01 0.96
× 𝑆 𝑛 +
𝑌
𝐶
𝐴
3
−1
2

10. Example cont.
b) State the initial state matrix
A gym has three programs: Yoga, CrossFit and Aerobics. Initially, there are
100 people doing the Yoga Program, 200 people are in the CrossFit
Program and 300 people in the Aerobics program.
𝑆0 =
𝑌
𝐶
𝐴
100
200
300

11. Example cont.
c) Find the number of people doing each activity after
one and two weeks.
𝑆1 = 𝑇𝑆0 + 𝐵
𝑆1 =
0.94 0.05 0.02
0.02 0.94 0.02
0.04 0.01 0.96
×
100
200
300
+
3
−1
2
𝑆1 =
113
195
296
After one week there are 113 people at Yoga, 195 people at
CrossFit and 296 people at Aerobics.

12. Example cont.
c) Find the number of people doing each activity after one
and two weeks.
𝑆2 = 𝑇𝑆1 + 𝐵
𝑆2 =
0.94 0.05 0.02
0.02 0.94 0.02
0.04 0.01 0.96
×
113
195
296
+
3
−1
2
𝑆2 =
124.89
190.48
292.63
After two weeks there are 125 people at Yoga, 190 people at
CrossFit and 293 people at Aerobics. Notice that the total
number of people in the population is changing.